Conditional Risk Minimization with Side Information: A Tractable, Universal Optimal Transport Framework

ArXiv ID: 2509.23128 “View on arXiv”

Authors: Xinqiao Xie, Jonathan Yu-Meng Li

Abstract

Conditional risk minimization arises in high-stakes decisions where risk must be assessed in light of side information, such as stressed economic conditions, specific customer profiles, or other contextual covariates. Constructing reliable conditional distributions from limited data is notoriously difficult, motivating a series of optimal-transport-based proposals that address this uncertainty in a distributionally robust manner. Yet these approaches remain fragmented, each constrained by its own limitations: some rely on point estimates or restrictive structural assumptions, others apply only to narrow classes of risk measures, and their structural connections are unclear. We introduce a universal framework for distributionally robust conditional risk minimization, built on a novel union-ball formulation in optimal transport. This framework offers three key advantages: interpretability, by subsuming existing methods as special cases and revealing their deep structural links; tractability, by yielding convex reformulations for virtually all major risk functionals studied in the literature; and scalability, by supporting cutting-plane algorithms for large-scale conditional risk problems. Applications to portfolio optimization with rank-dependent expected utility highlight the practical effectiveness of the framework, with conditional models converging to optimal solutions where unconditional ones clearly do not.

Keywords: distributionally robust optimization, optimal transport, conditional risk minimization, convex reformulation, cutting-plane algorithms, Portfolio Optimization

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper presents a highly mathematical framework based on optimal transport and union-ball formulations, involving dense theoretical derivations and convex reformulations. It lacks empirical validation, backtests, or code; the practical application is a conceptual portfolio optimization example without reported performance metrics.

  flowchart TD
    A["Research Goal: Universal Framework for<br>Distributionally Robust Conditional Risk Minimization"] --> B["Novel Methodology: Union-Ball<br>Optimal Transport Formulation"]
    B --> C["Data Input: Limited Observed Data<br>& Contextual Covariates"]
    C --> D["Computational Process:<br>Convex Reformulation & Cutting-Plane Algorithms"]
    D --> E["Outcome: Unified Interpretability<br>of Existing Methods"]
    D --> F["Outcome: Tractable Solutions<br>for Major Risk Functionals"]
    D --> G["Outcome: Scalability for<br>Large-Scale Problems"]